Question

The potential energy of a particle moving along $x-$axis, under the action of conservative force, is given by $U=20+5\sin \left(4\pi x\right)$, where $U$ is in Joules and $x$ is in metres. Then which of the following is/are true?

• A. at $x=\frac{7}{8}\text{m}$, particle is at equilibrium
• B. at $x=\frac{7}{8}\text{m}$, particle is not at equilibrium.
• C. at $x=\frac{3}{8}\text{m}$, particle is at equilibrium.
• D. at $x=\frac{3}{8}\text{m}$, particle is not at equilibrium.

Answer 1

Answer: (A), (C)

The correct options are
A at $x=\frac{7}{8}\text{m}$, particle is at equilibrium
C at $x=\frac{3}{8}\text{m}$, particle is at equilibrium.

Given: $U=20+5\sin \left(4\pi x\right)$
Conservative force, $F=-\frac{dU}{dx}$
$\Rightarrow F=-20\pi \cos \left(4\pi x\right)$

At $x=\frac{7}{8}\text{m}$:-
$F=20\pi \cos (4\pi .\frac{7}{8})=20\pi \cos \frac{7\pi }{2}$
$\therefore F=0$ at $x=\frac{7}{8}\text{m}$
($\because \cos \frac{(2n+1)\pi }{2}=0$)

Similarly, at $x=\frac{3}{8}\text{m}$:-
$F=20\pi \cos (4\pi .\frac{3}{8})=20\pi \cos \frac{3\pi }{2}=0$
$\therefore$ Equilibrium at $x=\frac{7}{8}\text{m}$ and $x=\frac{3}{8}\text{m}$